Download Improving Control Dynamics of PMSM Drive by Estimating Zero

http://dx.doi.org/10.5755/j01.eee.21.2.11507
ELEKTRONIKA IR ELEKTROTECHNIKA, ISSN 1392-1215, VOL. 21, NO. 2, 2015
Improving Control Dynamics of PMSM Drive
by Estimating Zero-Delay Current Value
Leszek Jarzebowicz1, Artur Opalinski1, Maciej Cisek1
Faculty of Electrical and Control Engineering, Gdansk University of Technology,
G. Narutowicza 11/12, 80–233 Gdansk, Poland
[email protected]
1
motor current value for the beginning of forthcoming
control cycle, i.e. zero-delay value. This reduces the control
loop step response rise time to a single control cycle, i.e. to
the shortest technically possible value, without hardware
changes and with negligible processing overhead.
1Abstract—Dynamic performance of current control loop
still remains crucial for position-, speed-, and torque-controlled
drives. In the study, a current loop solution has been designed
for field oriented control of permanent magnet synchronous
motors (PMSM). It enhances typical PI controller with an
estimator of zero-delay current (ZDC) value. The ZDC
estimation allows for selecting substantially higher controller
gain. It reduces control loop step response rise time to a single
control cycle, which is the shortest technically possible value,
while avoiding overshoot. The method does not require any
hardware changes and it needs only negligible processing
overhead. Both simulations made and experimental results
obtained in the study have proved the effectiveness of the
proposed solution.
II. APPROACHES TO IMPROVEMENT OF CURRENT CONTROL
LOOP PERFORMANCE
Recent works related to current control loop performance
can be classified into three groups. The first group aims at
improving current measurement accuracy. The second
proposes to improve the dynamic properties of current loop
by increasing the rate of control algorithm response without
increasing the pulse width modulation (PWM) carrier
frequency. The last one is focused on predicting the control
variables for the forthcoming control cycles.
Impact on the control performance of electric drive from
scaling and offset errors in current measurement is analysed
by Kim et al. [5]. Jarzebowicz analyses the errors resulting
from transformation of sampled phase currents into rotating
coordinate frame [6]. Both papers contain methods for
compensating systematic errors which improve general
performance of current control.
Bocker and Buchholz prove that updating the PWM
generator with a ratio of 8 to 16 may significantly improve
control bandwidth [7]. Typical controllers allow for single
or double update of reference voltage in a PWM cycle [8].
Therefore, the proposed solution requires an extraordinary
PWM generator.
A predictive approach to field oriented current control
algorithm in electric drives is proposed by Cortes et al. [9].
Predictive current controller calculates future behaviour of
the system based on a model and a set of possible actuations
for the horizon of two forthcoming control cycles. Similar
method using Smith predictor is applied to an inverter
operating in active power filter by Zhou and Liu [4]. Both
approaches increase computational complexity and share
dependency on using exact system parameters, inherent to
model-based prediction. In turn, Anuchin and Kozachenko
propose to extend PI controller with a current predictor
which uses oversampling and digital filtering to calculate
the zero-delay value of DC motor armature current [10].
The aim of this work is to indicate an algorithm for
estimating zero-delay current (ZDC) values for PMSM drive
and to investigate how this estimation will influence the
dynamic performance of PI-based current control loop.
Index Terms—Control performance, permanent magnet
motors, variable speed drives, synchronous sampling.
I. INTRODUCTION
Dynamic properties of electric drives are crucial to many
industrial applications. The most demanding devices are
typically fitted with permanent magnet synchronous motors
(PMSM), which allow for very fast torque and speed
response due to their low stator inductances and low
moment of inertia, respectively.
Control structure of electric drive consists of those: an
inner current control loop and of outer speed and position
loops, optionally. Some recent efforts aim at improving the
speed and position control algorithms [1]–[3]. However,
dynamic response of the drive is ultimately limited by the
properties of the inner control loop. Therefore, the dynamic
performance of the current control loop remains crucial for
position-, speed-, and torque-controlled drives.
A typical current control loop is based on proportionalintegral (PI) controller, highly appreciated in industry due to
its simplicity and reliability. Alternative approaches, e.g.
predictive controllers, while being superior to PI-based
solutions in terms of dynamic properties, are substantially
more difficult to implement and require dedicated efforts to
reduce their dependency on inaccuracy and variability of
pre-determined drive parameters [4].
The work described in this paper addresses the gap
between PI and alternative controllers, to find balance
between implementation complexity and dynamic
performance. The proposed current loop solution enhances
the capabilities of typical PI controller by an estimation of
Manuscript received December 23, 2014; accepted March 16, 2015.
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ELEKTRONIKA IR ELEKTROTECHNIKA, ISSN 1392-1215, VOL. 21, NO. 2, 2015
The structure for field oriented control (FOC) of PMSM
is shown in Fig. 1. The motor currents are controlled in d-q
rotating reference frame [11]. For motors with surface
mounted permanent magnets the value of current id is forced
to zero. The reference value of current iq is set according to
the required torque. Both currents are regulated by PI
controllers fitted with decoupling and electromotive force
(EMF) compensation block [12].
controller in PMSM drive FOC with linear extrapolation to
estimate ZDC value from synchronous sampling; a method
for selecting ZDC-enhanced PI controller gain; a timedomain model of digitally controlled drive to verify the
effectiveness of the enhancement; implementing the ZDC PI
controller; an experimental validation procedure; an
assessment of increase in control algorithm computational
complexity due to the enhancement.
III. ESTIMATING ZERO-DELAY CURRENT
The fundamental current component for a single PWM
cycle can be well approximated by linear dependency
[13], [14]. Thus the zero-delay current value i(tV[k]) for the
kth control cycle can be estimated by



 

i tV [k ]  2  i t P[k 1]  i tV [k 1] .
(1)
Estimation requires sampling frequency being only twice
the PWM frequency. The last current measurement i(tP[k-1])
used by (1) takes place in the midpoint of (k–1)th PWM
cycle, leaving half of cycle for control algorithm execution.
As measurements are carried out synchronously with PWM
carrier, no signal filtering is required to isolate the
fundamental current component.
The impact of the proposed ZDC estimation on the
dynamic performance of PMSM drive is investigated below
by the analysis of iq current step response. A mixed
continuous-discrete model of the drive was implemented in
MATLAB-Simulink. The general view of the model
consisting of PMSM and digital controller is in Fig. 3.
Fig. 1. General structure of FOC applied to PMSM drive.
Motor current contains ripple component induced by
modulated voltage (Fig. 2). The frequencies of the ripples
are much higher than the control bandwidth, thus controller
is capable to influence only fundamental current component.
This component should be extracted by control feedback in
order to provide stable and accurate torque control. A
convenient solution to this problem was reported by Blasko
et al. [13]. The authors proved that instantaneous current,
when sampled at the mid-points of passive inverter states,
corresponds to the fundamental component. This method of
sampling currents simultaneously with valleys or peaks of
PWM carrier is referred to as synchronous sampling.
PWM voltage sequence for kth control cycle has to be set
up before this cycle starts, i.e. before the tV[k] instant in
Fig. 2. Therefore the PWM update must be computed using
the current value sampled either at tV[k-1] or tP[k-1] instants.
This causes a delay of T or T/2, respectively, between the
current measurement instant and PWM update instant. The
delay reduces the dynamics of current control.
Fig. 3. Top-level view of a model of digitally controlled PMSM drive
implemented in Simulink.
The PMSM model relies on standard continuous-time
equations [2], [15]. The controller is modelled with
triggered subsystems executed at the midpoints or at the
endpoints of control cycles. This corresponds to instants of
signal sampling and to instants of inverter output update.
The implemented control structure is as shown previously in
Fig. 1.
To assess fundamental component rise time, one has to
get rid of the ripple component of motor currents resulting
from changes of inverter states in each PWM cycle. These
states depend on such factors as rotor position or
instantaneous DC-bus voltage. To make the rise time
Fig. 2. Selected waveforms in PMSM drive supplied by space vector
PWM-controlled voltage source inverter.
The contribution of this work consists of: enhancing PI
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measurements independent from such uncontrolled factors,
the modulated voltage was replaced in the model by voltage
mean value calculated for each control cycle. As a result, the
motor currents consist of only fundamental component.
The scope of the analysis includes the proposed ZDC
approach and two typical current sampling scenarios, i.e.
sampling at tV[k-1] or tP[k-1] instants. The model was set up to
match laboratory drive parameters listed in Table I. The
controller I-term reset time was set to TI = Lq/Rs which
follows the common practice of pole-zero-cancelation [8].
The controller gain KP was adjusted by experimenting
individually for each case to obtain 5 per cent overshoot.
Quantitative comparison of control loop dynamic
performance is based on measuring rise time T90%, defined
as the time required for the response to rise from 0 % to
90 % of its final value. The results of simulation including
individual controller gains KP and rise times T90% are given
in Fig. 4. The comparison shows the superiority of using
ZDC approach over typical synchronous sampling scenarios.
ZDC estimation enables for setting substantially higher
controller gains to obtain the same overshoot. Therefore the
rise time is reduced over 3 or over 2.5 times when compared
to sampling at valleys or peaks, respectively.
Changes of q-axis current in PMSM can be calculated as
[16], [17]
d iq
dt



1
 Rs iq  Ld pm id  pm f  uq .
Lq
(2)
The term uq_dec = – Ldpmid – pmf is computed by the
decoupling and EMF-compensation block and bypasses the
PI controller [12], [15]. The voltage drop uR = – Rsiq has a
negligible impact on dynamic behavior. Therefore (2) can be
simplified in terms of controller activity
d iq
dt

uq _ PI
Lq
(3)
,
and reformulated into discrete time-domain
iq[ k ]
T

uq _ PI [ k ]
Lq
(4)
.
Following the requirements of desired dynamic response
the current has to change by iq[k] = iq(tv[k+1]) – iq_ref(tv[k])
during time T of PWM cycle. Thus the gain Kp of the
controller should be selected to satisfy
KP 
uq _ PI [ k ]
iq[ k ]

Lq
T
.
(5)
The step response with controller gain set according to (5)
was investigated by simulation (Fig. 5). Under step
command the controller sets the output voltage uq_PI to a
value which cancels the error at the endpoint of the nearest
control cycle (Fig. 5(a)). The decoupling and EMFcompensation function produces the uq_dec component which
removes the influence of motor speed on controller activity.
Therefore, the control performed by the PI controller is not
affected by rotor speed as long as maximum inverter output
voltage is not exceeded (Fig. 5(b)).
Fig. 4. Comparison of step response rise times for ZDC approach and
typical synchronous sampling scenarios.
TABLE I. PMSM DRIVE PARAMETERS.
Parameter
Value
PWM carrier frequency f = 1/T
10 kHz
Rated phase current In
10 A
Rated DC-bus voltage UDCn
216 V
120 rad/s
Rated speed (mechanical) mn
d-axis component of stator inductance Ld
0.9 mH
q-axis component of stator inductance Lq
1.05 mH
75 mWb
Flux linkage due to the rotor magnets f
Number of pole pairs p
9
IV. SELECTING CONTROLLER GAIN
In typical synchronous sampling scenarios the response
time is a trade-off with respect to overshoot. The proposed
ZDC approach allows for avoiding overshoot without
increasing the rise time considered as number of control
cycles. The response may still be achieved in a single
control cycle if only the inverter output voltage does not
reach the limit resulting from DC bus voltage.
The fastest digital control response would bring the
current to its commanded value at the endpoint of the
forthcoming cycle, i.e. iq(tv[k+1]) = iq_ref(tv[k]). Simultaneously,
such a response would be featured by null overshoot. The
following analysis aims to select the controller gain KP to
obtain such a response.
a)
b)
c)
Fig. 5. Step responses in ZDC-enhanced controller for gain
set
accordingly
to
(5)
and
for
various
rotor
speeds:
a) m = 0; b) m = 0.25 p.u.; c) m = 0.5 p.u.
For relatively high speeds and high current steps the sum
of responses from the controller and EMF-compensation
function uq_ref = uq_PI + uq_dec may exceed the maximum
inverter output voltage. In such a case, the response takes
more than one control cycle (Fig. 5(c)). Nevertheless,
controller still ensures the shortest technically possible
response, considering voltage limitation.
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ELEKTRONIKA IR ELEKTROTECHNIKA, ISSN 1392-1215, VOL. 21, NO. 2, 2015
time is negligible.
In future works, more formal analysis of ZDC control
loop dynamics, to allow for comparing ZDC to competitive
solutions in terms of bandwidth, should be aimed.
V. EXPERIMENTAL VALIDATION
The ZDC estimator was implemented in the controller of
a laboratory drive with parameters listed in Table I. The
control algorithm from Fig. 1 is performed by
TMS320F2812 digital signal processor (DSP) running at
120 MHz. Executing ZDC formula (1) for 16-bit integer
variables of iA and iB currents takes 6 cycles of CPU clock,
i.e. 50 ns. This constitutes a negligible fraction of the total
control algorithm execution time per cycle which equals
35 μs.
The iq current is only a variable of control algorithm
which is calculated based on iA and iB motor currents, so it
cannot be measured directly. Therefore the experiment was
performed with the rotor of PMSM fixed at electrical angle
of 3/2 rad. In this position, the q-axis is aligned to A-axis,
hence iq = iA. This allows for measuring the iq current
indirectly, by sensing motor phase current iA.
The iA current waveform upon a step change of iq
reference value was recorded using LEM LTS-15NP
transducer and a digital oscilloscope (Fig. 6). The recording
is supplemented with waveforms of controller digital
outputs which indicate sample-and-hold instances and
control algorithm execution duration. The current rises from
zero to the nominal value in a single control cycle.
Substantial ripples visible in the current waveform are
caused by PWM switching. The digital controller only
controls the fundamental current component (dotted line in
Fig. 6) which is featured by null overshoot. This proves that
ZDC method offers single-cycle rise time with no overshoot
if the inverter output voltage does not reach the limit
resulting from DC bus voltage.
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VI. CONCLUSIONS
The ZDC approach demonstrates its ability to
substantially improve the dynamic properties of PMSM
current loop, while only marginally increasing PI controller
computational
complexity.
Both
simulation
and
experimental results verify the effectiveness of the proposed
solution.
[10]
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Fig. 6. Oscilloscope-registered
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iq
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